Question

Cos^2x + cos2x = 1 Решите, пожалуйста.

Answer 1

$cos^2x+cos2x=1\\\\cos^2x=1-cos2x\\\\\star \; \; cos2x=cos^2x-sin^2x=cos^2x-(1-cos^2x)=2cos^2x-1\; \Rightarrow \\\\2cos^2x=1+cos2x\; \; \to\; \; cos^2x= \frac{1+cos2x}{2} \; \; \star \\\\\\ \frac{1+cos2x}{2}=1-cos2x\\\\1+cos2x=2-2cos2x\\\\3cos2x=1\\\\cos2x= \frac{1}{3}\\\\ 2x=\pm arccos \frac{1}{3}+2\pi n,\; n\in Z\\\\\underline {x=\pm \frac{1}{2}\cdot arccos \frac{1}{3}+\pi n,\; n\in Z}$